The Annals of Probability

Normal approximation under local dependence

Louis H. Y. Chen and Qi-Man Shao

Full-text: Open access

Abstract

We establish both uniform and nonuniform error bounds of the Berry–Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs couple Stein’s method with the concentration inequality approach.

Article information

Source
Ann. Probab., Volume 32, Number 3 (2004), 1985-2028.

Dates
First available in Project Euclid: 14 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1089808417

Digital Object Identifier
doi:10.1214/009117904000000450

Mathematical Reviews number (MathSciNet)
MR2073183

Zentralblatt MATH identifier
1048.60020

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G60: Random fields

Keywords
Stein’s method normal approximation local dependence concentration inequality uniform Berry–Esseen bound nonuniform Berry–Esseen bound random field

Citation

Chen, Louis H. Y.; Shao, Qi-Man. Normal approximation under local dependence. Ann. Probab. 32 (2004), no. 3, 1985--2028. doi:10.1214/009117904000000450. https://projecteuclid.org/euclid.aop/1089808417


Export citation

References

  • Baldi, P. and Rinott, Y. (1989). On normal approximation of distributions in terms of dependency graph. Ann. Probab. 17 1646--1650.
  • Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics. Papers in Honor of Samuel Karlin (T. W. Anderson, K. B. Athreya and D. L. Iglehart, eds.) 59--81. Academic Press, Boston.
  • Berry, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variables. Trans. Amer. Math. Soc. 49 122--136.
  • Bikelis, A. (1966). Estimates of the remainder in the central limit theorem. Litovsk. Mat. Sb. 6 323--346.
  • Chen, L. H. Y. (1978). Two central limit problems for dependent random variables. Z. Wahrsch. Verw. Gebiete 43 223--243.
  • Chen, L. H. Y. (1986). The rate of convergence in a central limit theorem for dependent random variables with arbitrary index set. IMA Preprint Series #243, Univ. Minnesota.
  • Chen, L. H. Y. (1998). Stein's method: Some perspectives with applications. Probability Towards 2000. Lecture Notes in Statist. 128 97--122. Springer, Berlin.
  • Chen, L. H. Y. and Shao, Q. M. (2001). A non-uniform Berry--Esseen bound via Stein's method. Probab. Theory Related Fields 120 236--254.
  • Dasgupta, R. (1992). Nonuniform speed of convergence to normality for some stationary $m$-dependent processes. Calcutta Statist. Assoc. Bull. 42 149--162.
  • Dembo, A. and Rinott, Y. (1996). Some examples of normal approximations by Stein's method. In Random Discrete Structures (D. Aldous and R. Pemantle, eds.) 25--44. Springer, New York.
  • Erickson, R. V. (1974). $L_1$ bounds for asymptotic normality of $m$-dependent sums using Stein's technique. Ann. Probab. 2 522--529.
  • Esseen, C. G. (1945). Fourier analysis of distribution functions: A mathematical study of the Laplace--Gaussian law. Acta Math. 77 1--125.
  • Esseen, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrsch. Verw. Gebiete 9 290--308.
  • Heinrich, L. (1984). Nonuniform estimates and asymptotic expansions of the remainder in the central limit theorem for $m$-dependent random variables. Math. Nachr. 115 7--20.
  • Ho, S.-T. and Chen, L. H. Y. (1978). An $L_p$ bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231--249.
  • Nagaev, S. V. (1965). Some limit theorems for large deviations. Theory Probab. Appl. 10 214--235.
  • Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Varaibles. Clarendon Press, Oxford.
  • Prakasa Rao, B. L. S. (1981). A nonuniform estimate of the rate of convergence in the central limit theorem for $m$-dependent random fields. Z. Wahrsch. Verw. Gebiete 58 247--256.
  • Rinott, Y. (1994). On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55 135--143.
  • Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with $n\sp-1/2\log n$ rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333--350.
  • Shergin, V. V. (1979). On the convergence rate in the central limit theorem for $m$-dependent random variables. Theory Probab. Appl. 24 782--796.
  • Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 583--602. Univ. California Press, Berkeley.
  • Stein, C. (1986). Approximation Computation of Expectations. IMS, Hayward, CA.
  • Sunklodas, J. (1999). A lower bound for the rate of convergence in the central limit theorem for $m$-dependent random fields. Theory Probab. Appl. 43 162--169.
  • Tihomirov, A. N. (1980). Convergence rate in the central limit theorem for weakly dependent random variables. Theory Probab. Appl. 25 800--818.